The Antitriangular Factorization of Saddle Point Matrices

Mastronardi and Van Dooren [this journal, 34 (2013) pp. 173–196] recently introduced the block antitriangular (“Batman”) decomposition for symmetric indefinite matrices. Here we show the simplification of this factorisation for saddle point matrices and demonstrate how it represents the common nullspace method. We show that rank-1 updates to the saddle point matrix can be easily incorporated into the factorisation and give bounds on the eigenvalues of matrices important in saddle point theory. We show the relation of this factorisation to constraint preconditioning and how it transforms but preserves the structure of block diagonal and block triangular preconditioners.